Theorem u3lem5 763

Description: Lemma for unified implication study. (Contributed by NM, 24-Dec-1997.)

Assertion

Ref	Expression
u3lem5	(a →3 (a →3 b)) = (a⊥ ∪ b)

Proof of Theorem u3lem5

Step	Hyp	Ref	Expression
1		comi31 508	. . 3 a C (a →3 b)
2	1	u3lemc4 703	. 2 (a →3 (a →3 b)) = (a⊥ ∪ (a →3 b))
3		ax-a2 31	. . 3 (a⊥ ∪ (a →3 b)) = ((a →3 b) ∪ a⊥ )
4		u3lemona 627	. . 3 ((a →3 b) ∪ a⊥ ) = (a⊥ ∪ b)
5	3, 4	ax-r2 36	. 2 (a⊥ ∪ (a →3 b)) = (a⊥ ∪ b)
6	2, 5	ax-r2 36	1 (a →3 (a →3 b)) = (a⊥ ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u3lem6  767  u3lem14mp  794